It is by now well known, that at thermal equilibrium, the dynamics of multimoded optical configurations eventually settle into a Rayleigh-Jeans (RJ) distribution in the presence of weak nonlinearities. Yet, in spite of intense research, a general methodology to quantify and predict the complete statistical response of optical microcanonical settings remains elusive. Toward this end, we here develop a universal theory of fluctuation statistics for nonlinear multimode optical systems. Our results reveal a transition from narrow quasi-Lorentzian statistics in the low temperature regime, to an exponential distribution at higher temperatures. We show that this peculiar morphing of the underlying photostatistics is unique to microcanonical systems and has no analog in grand canonical configurations. A comparison between direct phase space integration and numerical simulations shows excellent agreement with our theoretical results, providing a testament to ergodicity even in nonlinear systems with very few modes. Through our formal methodology, we are able to accurately quantify the nonlinear equilibria of small sized configurations, demonstrating a strong deviation from RJ statistics and an equilibrium response that defies equipartition at infinite temperatures.

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• Classical statistical mechanics
• Nonlinear optics